Discontinuities in self-affine functions lead to multiaffinity.

Many systems of both theoretical and applied interest display multiaffine scaling at small length or time scales. We demonstrate analytically and numerically that when vertical discontinuities are introduced into a self-affine function, the function becomes multiaffine. The discontinuities may correspond to surface overhangs or some source of discontinuous noise. Two functions are numerically examined with different distributions of discontinuities. The multiaffinity is shown to arise simply from the function of discontinuities, and the analytic scaling form at small scales for the function of discontinuities is derived and compared to numerical results.